From \(\operatorname{\mathbf {\ell }}_p\)  distances to LPIPS.
One of the main disadvantages of \(\operatorname{\mathbf {\ell }}_p\) -norms is that they are very sensitive under simple transformations such as rotations or translations [1]}. One possible solution is to consider perceptual distancesNot necessarily distances in a strict mathematical sense that assumes a certain set of axioms to hold. which capture these invariances better such as the learned perceptual image patch similarity (LPIPS) distance introduced in [2]} and which is based on the activations of a convolutional network. The LPIPS distance is formally defined as
\(\operatorname{d_{\text{LPIPS}}}(x,x^{\prime })^2= \sum _{l=1}^L \alpha _l\Vert \phi _l(x)-\phi _l(x^{\prime })\Vert _2^2,\) 
